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Polynomial functions are mathematical expressions composed of variables raised to whole number powers and their coefficients. In essence, a polynomial is an addition of terms, which are products of a constant and a variable raised to a non-negative integer exponent. For instance, the polynomial function in the exercise,

\[P(x) = 4x^4 - 6x^2 + 5\]

has terms with degrees of 4, 2, and 0. The highest exponent term defines the degree of the polynomial—in this case, degree four. A notable feature of polynomial functions is that they are continuous and smooth. This means that their graphs do not have holes, jumps, or sharp turns. Understanding polynomial functions is crucial for analyzing their behavior, such as identifying the turning points and intercepts, and for solving calculus problems such as finding rates of change.

Synthetic division is a shorthand method for dividing a polynomial by a binomial of the form \(x-c\). It is generally used when the divisor is linear. Compared to the traditional long division, synthetic division is much faster and simpler to carry out. The process involves writing down only the coefficients of the polynomial, excluding variables and exponents, and then performing a series of multiplication and addition operations.

Despite not being used directly in the exercise, it's important to grasp that synthetic division could have been employed as an alternative method to find the remainder when \(P(x)\) is divided by \(x-c\), which is precisely what the Remainder Theorem states the value of \(P(c)\) would be. Students often use synthetic division because it efficiently reveals not only the remainder but also the quotient of the division.

The evaluation of \(P(c)\), according to the Remainder Theorem, is the remainder one would get if they divided the polynomial \(P(x)\) by \(x-c\). To find \(P(c)\), you simply replace the variable \(x\) with the constant \(c\) throughout the polynomial and then perform the necessary arithmetic operations. This is precisely what was done in the exercise. The value of \(c\) was given as -2, resulting in the evaluation

\(P(-2) = 4*(-2)^4 - 6*(-2)^2 + 5\).

Simplifying this gave \(P(-2) = 64 - 24 + 5\), which equals 45. This means that if we were to divide the given polynomial by \(x - (-2)\) or \(x + 2\), the remainder of that division would be 45. This value can be used in various contexts, such as determining zeros of polynomials or constructing polynomial graphs.